Optimal Control of 1D Semilinear Heat Equations with Moment-SOS Relaxations

TL;DR

This paper introduces a moment-SOS relaxation approach for optimal boundary control of 1D semilinear heat equations, enabling the extraction of nonlinear controllers and demonstrating improved performance over linear-quadratic controllers.

Contribution

It extends moment-based PDE control methods to nonlinear cases with quadratic costs and develops a new extraction technique for nonlinear boundary controllers.

Findings

Method successfully computes nonlinear boundary controllers.

Numerical validation shows improved control performance.

Comparison indicates advantages over linear-quadratic controllers.

Abstract

We use moment-SOS (Sum Of Squares) relaxations to address the optimal control problem of the 1D heat equation perturbed with a nonlinear term. We extend the current framework of moment-based optimal control of PDEs to consider a quadratic cost on the control. We develop a new method to extract a nonlinear controller from approximate moments of the solution. The control law acts on the boundary of the domain and depends on the solution over the whole domain. Our method is validated numerically and compared to a linear-quadratic controller.